Quantifying the brain's sheet structure with normalized convolution

Abstract

The hypothesis that brain pathways form 2D sheet-like structures layered in 3D as “pages of a book” has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of “path neighborhoods” reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications of the sheet structure hypothesis for our understanding of brain structure and development, it is still considered controversial by many for lack of quantitative analysis. A means to quantify sheet structure is therefore necessary to reliably investigate its occurrence in the brain. Previous work has proposed the Lie bracket as a quantitative indicator of sheet structure, which could be computed by reconstructing path neighborhoods from the peak orientations of dMRI orientation density functions. Robust estimation of the Lie bracket, however, is challenging due to high noise levels and missing peak orientations. We propose a novel method to estimate the Lie bracket that does not involve the reconstruction of path neighborhoods with tractography. This method requires the computation of derivatives of the fiber peak orientations, for which we adopt an approach called normalized convolution. With simulations and experimental data we show that the new approach is more robust with respect to missing peaks and noise. We also demonstrate that the method is able to quantify to what extent sheet structure is supported for dMRI data of different species, acquired with different scanners, diffusion weightings, dMRI sampling schemes, and spatial resolutions. The proposed method can also be used with directional data derived from other techniques than dMRI, which will facilitate further validation of the existence of sheet structure.

The Lamellar Structure of the Brain Fiber Pathways

“…The original conclusions of Wedeen et al (2012a) were based a non-quantitative visual assessment of intervoxel fiber relationships. In a recent series of abstracts, Tax and colleagues entered this debate by aiming to more quantitatively describe sheet structures from intervoxel structure using well-known differential geometric methods (Tax et al, 2013, 2014, 2015; Tax, Haije, et al, 2016; Tax, Westin, et al, 2016). Unfortunately, their approach is incapacitated by their reliance on standard DTI analysis methods which are incapable of accurately discerning intravoxel distributions, thus requiring, for example, the invocation of artificial methods to account for the missing peaks in their characterization of the local diffusion by an fiber orientation distribution function (fODF) (Tax, Westin, et al, 2016).…”

“…Moreover, it is claimed that the Cartesian q -space sampling used in their implementation of DSI presents a lower risk of bias and is thus a more conservative choice of methods when microstructure is uncertain (Wedeen et al, 2012b). More recently, in a sequence of workshop presentations (Tax et al, 2013, 2014, 2015; Tax, Haije, et al, 2016; Tax, Westin, et al, 2016) several quantitative techniques were attempted to evaluate the existence of sheet structure, but no convincing arguments were provided.…”

Effects of Inaccurate Response Function Calibration on Characteristics of the Fiber Orientation Distribution in Diffusion MRI

“…We clustered the peak directions to make sure that we are always comparing the angular deviations between the simulated fiber orientation and the FOD peak orientation most closely aligned to that orientation. Like in other studies [16,32,33] that compare axial and radial diffusion characteristics, we also included an angular threshold (e.g., cos ( θ ) > 0.7, which means approximately θ < 45°) to make sure the correct peaks were being extracted for further evaluations.…”

Advances in computational and statistical diffusion MRI

“…151 The advances described in this article represent crucial steps forward for dMRI computational analysis. However, important basic research is under way that may change our assumptions about the brain's microstructure and connectivity: current theories about the organization of axons into sheets [152][153][154][155] and the possibility that axons take sharp turns or branch at 90 degrees throughout the brain 156 have important implications for the future of tractography methods and microstructure models. In fact, the true three-dimensional morphology of neurons is not yet known and it is currently an active field of study.…”